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# MATH 142 Test 1 STUDY GUIDE MATH 142

USC

GPA 3.7

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This 7 page Study Guide was uploaded by Abbey McCoy on Thursday February 4, 2016. The Study Guide belongs to MATH 142 at University of South Carolina taught by Dr. Zeimke in Spring 2016. Since its upload, it has received 61 views. For similar materials see Calculus II in Applied Mathematics at University of South Carolina.

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Date Created: 02/04/16

Test Date: February 9 MATH 142 Test 1 Study Guide 5.4: The Fundamental Theorem of Calculus- o Part 1: if f is continuous on the interval [a,b], then… ???? ???? ???? = ∫ ???? ???? ???????? ???? …is continuous on [a,b] and is differentiable on (a,b), and its derivative is f(x). ′ ???? ???? ???? ???? = ????????∫???? ???? ???? ???????? = ???? ????) o Part 2: if f is continuous on [a,b], and F is an antiderivative of f (that is, F’(x) = f(x)), then… ???? ∫ ???? ???? ???????? = ???? ???? − ???? ???? ( ) ???? 5.5: Indefinite Integrals and the Substitution Method- o Indefinite Integrals: the integral of a function with respect to x is the set of all antiderivatives f and is denoted by… ∫???? ???? ???????? o The Substitution Method: the chain rule backwards 2 −4 Ex:∫2???? ???? + 5 ) ???????? 2 ???? = ???? + 5 ???????? = 2???????????? −4 = ∫???? ???????? ????−3 = + ???? −3 1 = − (???? + 5 )−3+ ???? 3 5.6: Definite Integral Substitutions and the Area Between Curves- o If g’ is continuous on [a,b] and f is continuous on range of g(t)=u, then… Test Date: February 9 ???? ????(????) ∫ ????(???? ???? )???? ???? ???????? = ∫ ???? ???? ???????? ???? ????(????) 8.1: Using Basic Integration Formulas- ????+1 ∫???????????? = ???????? + ????,???? = ???????????? ???????????????????????? ∫???? ???????? = ???? + ????,???? ≠ −1 ???? + 1 ???????? ∫ = ???????? ???? + ???? ∫???? ???????? = ???? + ???? ???? ???????? ∫???? ???????? = + ????, ???? > 0,???? ≠ 1) ∫sin ???? ???????? = −cos ???? + ???? ln ????) 2 ∫cos ???? ???????? = sin ???? + ???? ∫sec ???? ???????? = tan ???? + ???? ∫csc ???? ???????? = −cot ???? + ???? ∫sec ???? tan ???? ???????? = sec ???? + ???? ∫csc ???? cot ???? ???????? = −csc ???? + ???? ∫tan ???? ???????? = ???????? sec ???? + ???? ∫cot ???? ???????? = ???????? sin ???? + ???? ∫sec ???? ???????? = ???????? sec ???? + tan(????) + ???? ???????? ???? ∫csc ???? ???????? = −???????? csc ???? + cot ???? + ???? ∫ = arcsin( ) + ???? √???? − ???? 2 ???? ???????? 1 ???? ???????? 1 ???? ∫ ???? + ???? 2= ????arctan( ???? + ???? ∫ ????√???? + ???? 2= ???? ????????????????????????|????| + ???? 8.2: Integration by Parts- ???? ′ ′ o ????????(???? ???? ???? ???? ) = ???? ???? ???? ???? + ???? ???? ???? ???? ( ) ???? ′ ′ ∫ ???????? (???? ???? ???? ???? )???????? = ∫???? ???? ???? ???? ???????? + ∫???? ???? ???? ???? ???????? = ???? ???? ????(????) ) ∫???? ???? ???? ???? ???????? = ???? ???? ???? ???? − ∫???? ???? ???? ???? ???????? ( ) ???? = ???? ???? ) ???? = ???? ????) ′ ′ ???????? = ???? ???? ???????? ???????? = ???? ???? ???????? Test Date: February 9 ∫???????????? = ???????? − ∫???????????? 8.3: Trigonometric Integrals- o Integrals of Powers of Sines and Cosines: ???? ( ) ???? ∫sin ???? cos (????)???????? Case 1: if m is odd, then ???? = 2???? + 1 for some integer k 2 2 Use the identity sin (????) = 1 − cos ???? to obtain… ???? sin (????) = sin2????+1(????) = sin2????(???? sin ????) 2 ???? = sin ????( )) sin ???? = 1 − cos ????( ))???? sin(????) Then use substitution with ???? = cos(????) Case 2: if m is even and n is odd, then ???? = 2???? + 1 for some integer k 2 2 Use the identity cos (????) = 1 − sin (????) to obtain… cos ???? ) 2????+1 = cos (????) = cos ????( )) ????cos ????) 2 ???? = 1 − sin ????( )) cos ????) Then use substitution with ???? = sin(????) Case 3: if both m and n are even Use these identities… 2( ) 1 − cos(2????) sin ???? = 2 1 − sin(2????) cos (????) = 2 cos ???? + ???? = cos ???? cos ???? − sin ???? sin ???? ( ) sin ???? + ???? = sin ???? sin ???? + cos ???? cos ???? ( ) 2 2 cos 2???? = cos ???? − sin ???? ( ) = cos ???? − 1 − cos ???? 2( )) = 2cos ???? − 1 Test Date: February 9h o Integrals of Powers of Tangent and Secant: When integrating tangent and secant with powers higher than 2, use the identities… tan ???? = sec ???? − 1) 2 2 sec ???? = tan ???? + 1) In order to reduce higher powers, use integration by parts. o Integrals of the following forms: ( ) ( ) ∫sin ???????? sin ???????? ???????? ∫sin ???????? cos ???????? ????????) ∫cos ???????? cos ???????? ???????? ) Use… 1 sin ???????? sin ???????? =) 2(cos( ???? − ???? ????) − cos( ???? + ???? ????)) 1 sin ???????? cos ???????? = (sin( ???? − ???? ????) + sin( ???? + ???? ????)) ) 2 1 cos ???????? cos ???????? =) 2 (cos( ???? − ???? ????) + cos( ???? + ???? ????)) 8.4: Trigonometric Substitution- o Use this method when the integrand involves expressions such as… √ ???? + ???? 2 √???? − ???? 2 √???? − ???? 2 o Substitute… ???? = sec ???? ) ???? = sin(????) ???? = tan ???? ) ???? ???? o sin(x) is invertible (can use arcsin(x)) on2 2 , ] ???? ???? o tan(x) is invertible on (−2 2) ???? ???? o sec(x) is invertible on 0 < ???? <2 if ???? ≥ 1 and 2 < ???? < ???? if ???? ≤ −1 8.5: Integration of Rational Functions by Partial Fractions- ???????????????????????????????????????? o Use when integrating a rational function ???????????????????????????????????????? ???? ????) o Partial Fraction Decomposition: assume ???? ???? = ???? ???? )here P and Q are polynomials and the degree of P is less than the degree of Q Test Date: February 9 4 Cases: 1. If Q(x) has distinct real roots and no irreducible factors, then… ???? ???? = ???? ???? + ???? )(???? ???? + ???? )…(???? ???? + ???? ) 1 1 2 2 ???? ???? Then… ???? ????) = ???? 1 + ???? 2 + ⋯+ ???????? ???? ????) ???? 1 + ???? 1 ???? 2 + ???? 2 ???????????? + ???????? 2. Q(x) has no irreducible factors but has repeated roots. Say (???????? + ????)???? is a factor, then the corresponding parts of the decomposition is given by… ????1 ???? 2 ???? ???? + + ⋯+ ???? ???????? + ???? ???????? + ???? (???????? + ????) 3. If Q(x) has an irreducible factor of the form ???????? + ???????? + ???? which doesn’t repeat, the corresponding form is then… ???????? + ???? 2 ???????? + ???????? + ???? 4. If Q(x) has an irreducible factor of the form ???????? + ???????? + ???? , the corresponding form is then… ???? 1 + ???? 1 ???? 2 + ???? 2 ???? ???? + ???? ???? ???????? + ???????? + ???? + ???????? + ???????? + ???? + ⋯+ ???????? + ???????? + ???? ???? ????) o When the degree of Q is less than the degree of???? ???? ), use long division, and then use partial fraction decomposition for the remainder 8.7: Numerical Integration- o Use numerical methods to find polynomials to help us integrate th Test Date: February 9 o Trapezoidal Rule: to approximate∫???????? ???? ????????, use T, where… ???? T=???????? (???? ????) + 2???? ????) + 2???? ???? ) + ⋯+ 2???? ???? ) ) 2 0 1 2 ???? where… ????−???? Δ???? = , 0 = ????, ????1= ???? + Δx, ???? 2 a + 2Δx,… , ???? = ???? ???? Similar to Riemann Sums ???? o Theorem: Let T be the trapezoidal approximation t∫???????? ???? ???????? with n steps and ???? let T be the error (i.????. ???? = ????∫???? ???? ???? ????????). If ????′′ is continuous and M is an upper bound for ????′′ on [a,b], then… 3 | | ???? ???? − ???? ) ???????? = 12???? 2 8.8: Improper Integrals- ???? o Type 1- if ∫1???? ???? ???????? exists for every ???? ≥ ????, then… ∞ ???? ∫ ???? ???? ???????? = lim ∫ ???? ???? ????????) ???? ????→∞ 1 provided the limit exists. ???? Similarly, ∫???? ???? ???? ???????? exists for every???? ≤ ????, then… ???? ???? ∫ ???? ???? ???????? = lim ∫ ???? ???? ????????( ) −∞ ????→−∞ ???? provided the limit exists. ∞ ???? If∫???? ???? ???? ???????? and −∞ ???? ???? ???????? exist for some a, then… ∞ ???? ???? ∫ ???? ???? ???????? = ∫ ???? ???? ???????? + ∫ ???? ???? ???????? ( ) −∞ −∞ −∞ o Type 2- if t is continuous on (a,b], and discontinuous at a, then… ???? ???? ∫ ???? ???? ???????? = lim ∫+???? ???? ????????) ???? ????→???? ???? If f is continuous on [a,b) and discontinuous at b, then… ???? ???? ∫ ???? ???? ???????? = lim ∫−???? ???? ????????) ???? ????→???? ???? Test Date: February 9 ???? If f is continuous on [a,b] except at c, where ???? < ???? < ????,∫???????? ???? ???????? ???? and ∫???????? ???? ???????? exist, then… ???? ???? ???? ∫ ???? ???? ???????? = ∫ ???? ???? ???????? + ∫ ???? ???? ????????( ) ???? ???? ???? o Direct Comparison Theorem: suppose f and g are continuous with 0 ≤ ????(????) ≤ ????(????) for all ???? ≥ ???? ∞ ∞ If∫???? ???? ???? ???????? is convergent, the∫???? ???? ???? ???????? is convergent ∞ ∞ If∫???? ???? ???? ???????? is divergent, the∫???? ???? ???? ???????? is divergent o Limit Comparison Test: if the positive functions f and g are continuous on [????,∞) and if lim???? ????)= ????, 0 < ???? < ∞, then ∫∞ ???? ???? ???????? and ∫∞ ???? ???? ???????? both converge ????→∞ ???? ????) ???? ???? or both diverge. Test Information: o PURCHASE AND BRING BLUE BOOK!!!! Size: 8.5x11 Can be found at the university bookstore o 9-10 questions Similar to homework questions Easy to medium difficulty level o No formula sheet provided Memorize all your formulas, and don’t forget trig identities! o See practice test online for format and practice questions 5.4-5.6 are not included on the practice test, so make sure you look up separate practice questions for those if you need to! Question 9 on the practice test will NOT be on our test o There will be partial credit More focused on your understanding of the concept than on your ability to do algebra Good luck everyone!

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